Seismic performance assessment of a precast segmental column bridge considering soil−structure interaction and depth-varying multi-support ground motion inputs

Abstract

The precast segmental column (PSC) plays a vital role in both the design of new bridges and the rehabilitation of existing ones. Previous studies of PSCs have primarily focused on their individual seismic behavior. However, research on the seismic performance of entire bridges supported by PSCs, particularly those incorporating soil−structure interaction (SSI), remains limited. Moreover, the amplification of earthquake waves as they propagate along the pile foundation to the surface, coupled with the coherency loss between earthquake motions at varying supports, may further impact the seismic responses of such bridges. This study systematically assesses the seismic performance of a PSC bridge (PSCB) supported by pile foundations considering the effects of SSI and depth-varying multi-support ground motions. Moreover, a benchmark bridge with the traditional monolithic column is also analyzed for comparison. The seismic fragility of bridges is calculated based on nonlinear time history analyses and joint probability density functions for both peak and residual responses. Parameter studies are conducted to reveal the influences of SSI, non-uniform excitation, and depth-varying earthquake loads on seismic performance assessments. This paper offers valuable insights for the reliable analysis of seismic response and fragility and the safety design of PSCB systems.

Graphical abstract

Keywords

precast segmental column bridge / soil−structure interaction / depth-varying multi-support ground motions / seismic performance / monolithic column bridge

1 Introduction

With the increase of structural service time, a lot of bridges around the world have been defined as “structurally deficient or functionally obsolete”. These bridges generally require immediate repair or reconstruction for safety, and the accelerated bridge construction (ABC) technique has emerged as an effective solution to address these needs [1]. The precast segmental column (PSC) has a promising future in ABC due to its reduced environmental consequences, minimal operation disruption, and reliable component quality [2,3]. Despite the evident advantages of PSC bridges (PSCBs) over traditional cast-in-place monolithic column bridges (MCBs), their applications have generally been limited to low seismicity regions because the seismic performances are still largely unknown. Consequently, it is important to carry out a comprehensive assessment of the seismic response and fragility for PSCBs.

To promote the wider adoption of PSCBs in areas prone to medium and high seismic activity, extensive studies on their seismic behavior have been carried out in recent decades, including theoretical analyses [4,5], experimental tests [6–9], and numerical simulations [10,11]. These studies underscored that PSCs exhibit two unique characteristics different from the traditional monolithic columns (MCs): 1) reduced residual displacement, owing to the self-centering ability offered by the prestressing tendons; and 2) lower energy dissipation (ED), as the PSC is not continuous but assembled from multiple segments, with the concrete in the column body remaining largely undamaged except at the joints. Moreover, some studies focusing on the seismic behavior of entire bridges with PSCs have also been conducted. Mantawy et al. [12] and Sideris et al. [13] studied the seismic response of PSCBs based on shaking-table tests, and the experimental results demonstrated the PSCB experienced negligible residual displacement across all ground motions. Zhao et al. [14] conducted numerical studies on different bridge systems, assessing the difference in seismic response between MCBs and PSCBs. A study by Li et al. [15] analyzed the seismic performance of a bridge with segmental ultra-high performance concrete piers and compared the seismic response and fragility of the bridge supported by different pier types (i.e., MC and PSC). Upadhyay and Pantelides [16] developed a design method for ABC bridges with unbonded post-tensioned columns using seismic fragility assessment. Li et al. [17] reported that the near-fault seismic motions have a significantly different impact on the seismic response of MCBs and PSCBs.

It is worth mentioning that all the aforementioned research works assumed that PSCBs were directly anchored to the surface, the soil−structure interaction (SSI) effect was roughly and irrationally ignored. In real-world engineering, bridge systems are often supported by multiple group pile foundations that penetrate multiple soil layers. Accordingly, the assumption of neglecting the SSI effect is not realistic in previous seismic response analyses of PSCBs. Numerous studies have been carried out to examine the impact of SSI on seismic response of various engineering structures [18–23]. These studies have demonstrated the SSI can dramatically influence the structural seismic performance assessment; this phenomenon is attributed that the SSI prominently affects whole structural dynamic characteristics [24,25]. However, the seismic response and fragility analyses of PSCBs considering the SSI effect have rarely been reported to date.

On the other hand, a common limitation found in the aforementioned investigations is that the spatial variations in earthquake motions were ignored during the seismic performance assessment of PSCBs. In reality, earthquake loadings at varying supports of bridges are inevitably different due to the wave passage, coherency loss, and local site effect [26–29]. To the authors’ knowledge, research works concerning the seismic analysis of PSCBs considering the spatial variations of seismic motions are rather limited. Only Zhao et al. [30] investigated the seismic performance of PSCBs subjected to spatially varying seismic excitations; however, their research merely considered the spatial variation of earthquake motions at site surface. Notably, ground motions at different site depths vary significantly because of the filtering and amplification effects of soil layers on seismic wave propagation [31,32]. Therefore, to accurately analyze the seismic response and fragility of PSCBs, depth-varying multi-support ground motions (DMGMs) should be adopted as input.

To address these key knowledge gaps, this paper proposes a seismic performance assessment method for PSCBs with SSI, using synthesized DMGMs as the earthquake input. The study explores the effects of SSI, non-uniform excitation, and depth-varying ground motions on the seismic performance assessment of PSCBs. Moreover, for comparison, the seismic performance of MCBs is also discussed. Specifically, Section 2 introduces a benchmark MCB and develops three-dimensional (3D) finite element (FE) models of both MCB and PSCB using the OpenSees platform. Section 3 outlines the SSI simulation using the p–y method and presents an artificial approach for generating DMGMs. The influences of the SSI, non-uniform excitation, and depth-varying ground motions on the seismic responses and fragilities of PSCBs are studied in Sections 4 and 5, respectively; moreover, the differences in seismic performance between MCB and PSCB are also discussed. Finally, Section 6 summarizes the main conclusions.

2 Analytical models

2.1 Benchmark monolithic column bridge and finite element modeling

A reinforced concrete rigid-frame bridge with four spans, part of a large-span bridge in China, is adopted as a case study. Fig.1 presents the bridge’s elevation and cross-sectional details. As shown, the bridge has a length of 45 + 55 + 55 + 45 = 200 m. It consists of a main girder and five pier columns, and the superstructure is supported by five large pile group foundations. The main girder is a concrete two-box design with a consistent cross section, and the area and moments of inertia of the section are given in Fig.1(b). The pier columns and piles have solid circular sections made of reinforced concrete. Fig.1(b) illustrates the features of the MC piers and piles cross sections. The pier columns and piles have heights of 11 and 24 m, respectively. Moreover, C40 concrete is employed for the bridge piers, while the concrete strength level of the pile foundations is C50 [33]. In particular, the main girder and pier columns of the bridge are rigidly connected without bearings.

The numerical models of the example bridge with varying types of pier columns are developed utilizing the OpenSees platform [34]. It should be noted that the MCB and PSCB have the same components except for the different pier column types. For brevity, this section focuses on the geometric characterization and modeling of the MCB, while the design and modeling details of the PSC will be presented in the next subsection. Specifically, since the main girder generally remains within the elastic range during seismic events, elastic beam−column elements are employed to simulate it. The MC piers are modeled adopted nonlinear beam−column elements, while the piles are represented by displacement-based beam column elements. The pile foundations and caps are connected using the Rigidlink command. The concrete material is simulated by Concrete02 considering the influence of stirrup constraint on concrete strength, and the longitudinal reinforcement material is modeled by Steel02 considering the isotropic strain hardening of the steel bar under repeated load. Moreover, the Rigidlink command is also used to connect the main girder and piers since they are fixed together.

2.2 Design and modeling of the precast segmental column

The segmental pier column should be properly designed for a fair comparison. Lee and Billington [35] studied the seismic behavior differences between MCBs and PSCBs, they proposed that both should have similar dimensions and comparable stiffness to enable a direct comparison of their seismic responses under earthquake excitations. In addition, Sakai and Mahin [36] found that precast post-tensioned concrete columns with similar stiffness and strength to the original columns can be achieved by reducing the number of longitudinal bars by nearly half and increasing the prestressed reinforcement, which also ensures the self-centering performance. Based on these recommendations, a series of pushover analyses (described in detail later) are carried out by changing the configuration of the longitudinal bars and prestressing tendons. The resulting design, including the structural form and cross-sectional reinforcement of the PSC, is shown in Fig.2(a) and Fig.2(b). The initial stress of the prestressing tendon is assigned to be 760 MPa.

For studies involving a large number of pushover analyses and seismic response calculations, it is critical to develop a reliable and accurate PSC FE model. This study employs the numerical simulation approach for PSC presented by Cai et al. [37] based on the OpenSees. Fig.2(a) illustrates the modeling details. Each precast reinforced concrete segment is modeled by employing one nonlinear beam column element. The concrete and longitudinal reinforcement of each segment are represented by the Concrete02 and Steel02 material models in OpenSees, respectively. The ZeroLengthSection element is employed to model the segment joint, with its fiber section similar to that of the PSC segment, except that the concrete fibers in the joint section have no tensile strength. The material properties of segment joints are defined by the Elastic-No Tension material in OpenSees. In addition, considering that the PSC joints primarily exhibit opening-closing behavior with negligible relative shear displacement when subjected to external loads [6,37], progressive damage associated with segment contact friction is not considered in this study. The EqualDOF command is used to define the constraints between adjacent segments. Moreover, the prestressing tendon is modeled using the CorotTruss element. It is assumed that the prestressing tendons remained fully engaged with the segments during seismic loading. Therefore, the RigidLink command is used to rigidly connect the two nodes of the CorotTruss element to the top and bottom nodes of the PSC segment, respectively. Based on the above method, pushover analyses are performed for both the designed PSC and MC, and the resulting hysteretic curves are presented in Fig.2(c). As shown, both pier column types have comparable stiffness and strength, and the PSC presents superior self-centering capability compared to the MC, which is attributed to the prestressing tendons. Another finding is that PSC has a lower ED capacity compared to the MC, these results are expected because the PSC is discontinuous and has less energy-dissipating bars.

3 Simulations of soil−structure interaction and depth-varying multi-support ground motions

3.1 Simulation of soil−structure interaction

To accurately conduct seismic response analyses, the soil spring method are used to simulate the SSI effect [18,20]. This approach models the pile−soil interaction using three mutually perpendicular soil springs: p–y, t–z, and Q–z. Fig.3 shows the schematic view of the pile foundations and soil connection in the FE model. As a supplement, Fig.3 also illustrates the location of each pile node in site soil layers; the soil distribution and soil parameters are also provided in detail. In this figure, G, ρ, ν’, ξ, n, Su, D_r, Ψ, and S_r signify the shear modulus, density, Poisson’s ratio of soil skeleton, damping ratio, porosity, undrained shear strength, relatively density, friction angle, and degree of saturation, respectively. Furthermore, each pile consists of 7 pile nodes with an equal spacing of 4 m, and thus 6 pile elements are defined. The zero-length elements are set at the 7 pile nodes, respectively, where one end of the element shares the degrees of freedom with the pile node, and the other end is subjected to earthquake loads. For each zero-length element, the PySimple1, TzSimple1, and QzSimple1 materials developed in Opensees [34] are employed to model the nonlinear relationship between soil pressure and displacement in the two horizontal directions (p–y), the vertical of pile body (t–z), and the vertical of pile end (Q–z).

Based on the calculation methods provided in API [38], the mechanical properties of soil springs at different positions of piles can be calculated, the detailed calculation schemes are not presented here for brevity. It is worth noting that the SSI between individual piles in the pile group foundation may influence each other, particularly for the p–y springs. Therefore, this study fully accounts for the pile group effect [39] when calculating the soil spring curves for the pile foundations. Fig.4 presents the p–y, t–z, and Q–z curves at varying pile foundation locations, with the corresponding positions indicated in Fig.1(a).

Based on the FE modeling approach and the SSI simulation details described above, modal analyses are conducted for both PSCB and MCB to verify the dynamic characteristics. Tab.1 summarizes the natural periods and corresponding mode shapes of the first five vibration modes for each FE model, including the PSCB with and without SSI and the MCB with SSI. The results show that in the first five modes, the bending vibration in the X and Y directions primarily governs the structural response. The SSI effect and pier type mainly influence the vibration periods, while their impact on the mode shapes is minimal. Compared to the model without SSI, the PSCB with SSI exhibits obvious longer vibration periods. This increase is due to the additional flexibility introduced by SSI, which changes the foundation boundary conditions from ideal rigid constraints to soil springs, thereby reducing the overall stiffness of the structural system. Furthermore, the vibration periods of PSCB and MCB are nearly identical. This similarity is attributed to the initial stiffness of the PSC, which is designed to be comparable to that of the MC, as shown in Fig.2(c).

3.2 Simulation of depth-varying multi-support ground motions

Traditional seismic response analysis of PSCBs usually uses spatially varying seismic excitations on the site surface (or even more unreasonable uniform seismic motions) as inputs. This inputs scenario typically ignores the variation characteristics of seismic motions along the pile foundation of bridges due to the site filtering and amplification effects, which may lead to inaccurate predictions of the structural seismic response. To reasonably analyze the seismic response of PSCBs, the 3D DMGMs in this study are generated using the classical spectral representation method combined with transfer functions and an empirical coherence loss function. This approach has been validated and applied in previous research, demonstrating its effectiveness in modeling spatial ground motions at a layered site [18,29,31,40]. For completeness, the simulation procedure of DMGMs is briefly described below.

According to the one-dimensional (1D) wave-propagation theory [41], the dynamic equilibrium equation for a site composed of multiple soil layers can be expressed as

(1)\left[S_{\mathrm{P}-\mathrm{S}\mathrm{V}}\right]\left\{u_{\mathrm{P}-\mathrm{S}\mathrm{V}}\right\}=\left\{P_{\mathrm{P}-\mathrm{S}\mathrm{V}}\right\}\mathrm{o}\mathrm{r}\left[S_{\mathrm{S}\mathrm{H}}\right]\left\{u_{\mathrm{S}\mathrm{H}}\right\}=\left\{P_{\mathrm{S}\mathrm{H}}\right\},

where \left[S_{\mathrm{P}-\mathrm{S}\mathrm{V}}\right], \left\{u_{\mathrm{P}-\mathrm{S}\mathrm{V}}\right\}, and \left\{P_{\mathrm{P}-\mathrm{S}\mathrm{V}}\right\} denote the total dynamic stiffness matrix (DSM), displacements vectors, and load vectors of in-plane coupled P and SV waves, respectively; \left[S_{\mathrm{S}\mathrm{H}}\right], \left\{u_{\mathrm{S}\mathrm{H}}\right\}, and \left\{P_{\mathrm{S}\mathrm{H}}\right\} denote the total DSM, displacements vectors, and load vectors of out-of-plane SH wave, respectively.

It is noteworthy that the significant influence of the water saturation in porous soil layers on seismic P-wave propagation was not considered in the 1D wave-propagation theory. To precisely calculate the ground motion transfer functions of the layered site, the approach presented by Yang and Sato [42] is utilized to compute the Poisson’s ratio (v) and P-wave velocity (V_{\mathrm{P}}) of porous soils

(2)v=\frac{\left({{\alpha }_{\mathrm{s}}}^2{M}_{\mathrm{s}}\right)/G+2v^{\prime }/\left(1-2v^{\prime }\right)}{2{{\alpha }_{\mathrm{s}}}^2{M}_{\mathrm{s}}/G+1/\left(1-2v^{\prime }\right)},

(3)V_{\mathrm{P}}=\sqrt{\frac{\lambda +2G+{{\alpha }_{\mathrm{s}}}^2{M}_{\mathrm{s}}}{\rho }},

where {\alpha }_{\mathrm{s}} and M_{\mathrm{s}} are the parameters associated with the soil grains and the porosity fluid pressure, respectively; \lambda is the Lame constant; G and v^{\prime } are the shear modulus and Poisson’s ratio of the soil skeleton, respectively; \rho represents the density of porous soil.

The DSM of each porous soil layer can be gained by substituting Eqs. (2) and (3) into the equation of DSM proposed by Wolf [41]. After that, the total DSM for the example site is obtained by combining the DSM of each soil layer with that of the base rock. Finally, the dynamic equilibrium equation for the site with multiple soil layers is formulated based on Eq. (1). The site transfer functions of seismic motions in the three directions (i.e., H_X\left(\mathrm{i}\omega \right), H_Y\left(\mathrm{i}\omega \right), and H_Z\left(\mathrm{i}\omega \right)) at any site location can be computed by solving Eq. (1) in the frequency domain. This study assumes the incident angles of seismic motions is 60°, the 3D seismic motion transfer functions at different site locations are plotted in Fig.5, with the corresponding point locations marked in Fig.1(a). As can be found in Fig.5, the site transfer functions at different locations have remarkable differences owing to the variations in the depths and properties of soil layers.

In addition, the filtered Tajimi–Kanai power spectral density (PSD) function is employed to simulate the PSD functions of seismic motions at the base rock (S_{\mathrm{b}\mathrm{r}}\left(\omega \right))

(4)S_{\mathrm{b}\mathrm{r}}\left(\omega \right)=\frac{{\omega }^4}{{\left({\omega }_{\mathrm{f}}^2-{\omega }^2\right)}^2+4{\xi }_{\mathrm{f}}^2{\omega }_{\mathrm{f}}^2{\omega }^2}\frac{{\omega }_{\mathrm{T}}^4+4{\xi }_{\mathrm{T}}^2{\omega }_{\mathrm{T}}^2{\omega }^2}{{\left({\omega }_{\mathrm{T}}^2-{\omega }^2\right)}^2+4{\xi }_{\mathrm{T}}^2{\omega }_{\mathrm{T}}^2{\omega }^2}{S}_0,

where {\omega }_{\mathrm{f}}=0.5\pi \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s} and {\omega }_{\mathrm{T}}=10\pi \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s} signify the central frequency of the high pass filter function and the Tajimi–Kanai PSD function, respectively; {\xi }_{\mathrm{f}}={\xi }_{\mathrm{T}}=0.6 denote the corresponding damping ratio; S_0= 0.0034{\mathrm{m}}^2/{\mathrm{s}}^3 is the scaling factor related to the seismic motion intensity.

To simulate the spatial effects between the ground motions at different positions of the base rock, the empirical coherence model presented by Hao et al. [27] is employed

(5)\begin{array}{rl}{\gamma }_{j^{\prime }{k}^{\prime }}\left(\mathrm{i}\omega ,d_{j^{\prime }{k}^{\prime }}\right)=& \mathrm{e}\mathrm{x}\mathrm{p}\left(-\beta d_{j^{\prime }{k}^{\prime }}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[-\alpha \left(\omega \right)\sqrt{d_{j^{\prime }{k}^{\prime }}}{\left(\omega /2\pi \right)}^2\right]\\ & \cdot \mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{i}\omega d_{j^{\prime }{k}^{\prime }}/v_{\mathrm{a}\mathrm{p}\mathrm{p}}\right),\end{array}

(6)\alpha \left(\omega \right)=\{\begin{array}{ll}{\displaystyle \frac{2\pi a}{\omega }}+{\displaystyle \frac{b\omega }{2\pi }}+c,& 0.134\mathrm{r}\mathrm{a}\mathrm{d}\le \omega \le 62.83\mathrm{r}\mathrm{a}\mathrm{d},\\ 0.1a+10b+c,& \omega >62.83\mathrm{r}\mathrm{a}\mathrm{d},\end{array}

where {\gamma }_{j^{\prime }{k}^{\prime }} is the coherency loss function; d_{j^{\prime }{k}^{\prime }} denotes the distance between any two points j^{\prime } and k^{\prime } at base rock; v_{\mathrm{a}\mathrm{p}\mathrm{p}} represents the apparent wave velocity; a, b, c, and \beta signify the coherence loss coefficients. In this study, their values are derived from seismic data recorded during Event 45 of the SMART-1 array [27]. The specific values are as follows: a=3.583\times {10}^{-3}, b=-1.811\times {10}^{-5}, c=1.177\times {10}^{-4}, and \beta =1.109\times {10}^{-4}.

Using the site transfer functions, the seismic motion PSD functions of base rock, and the coherence loss function, the auto PSD function (S_{jj}\left(\omega \right)) of any simulation point and the cross PSD function (S_{jk}\left(\omega \right)) between any two simulation points can be calculated as

(7)S_{jj}\left(\omega \right)=S_{\mathrm{b}\mathrm{r}}\left(\omega \right){\left|H_j\left(\mathrm{i}\omega \right)\right|}^2,

(8)S_{jk}\left(\omega \right)=S_{\mathrm{b}\mathrm{r}}\left(\omega \right){\gamma }_{j^{\prime }{k}^{\prime }}{H}_j\left(\mathrm{i}\omega \right){H_k\left(\mathrm{i}\omega \right)}^{\ast },

where H_j\left(\mathrm{i}\omega \right) and H_k\left(\mathrm{i}\omega \right) denote the transfer functions at the simulation points j and k, respectively, with ‘*’ denoting the complex conjugate.

Finally, based on the Eqs. (7) and (8), the PSD function matrix for DMGMs at the example site can be formulated and decomposed. The DMGMs at multiple points are then stochastically simulated using the classical spectral representation method [31]. A more detailed description of the DMGMs simulation methodology can be obtained from previous studies by Li et al. [31,40]. The 3D seismic motion acceleration time histories at varying positions of the pile foundations are illustrated in Fig.6, with the corresponding points marked in Fig.1(a). As illustrated, due to the local site effect and coherence loss between seismic motions at varying locations, the peak ground acceleration (PGA) and time history of seismic motions at different pile nodes have significant variability. These results emphasize the necessity of considering DMGMs in the seismic response and fragility analysis of PSCBs.

4 Seismic response analyses of the precast segmental column bridge

This section presents calculated seismic responses of the PSCB under different analytical cases, with a focus on the effects of SSI and seismic motion input types. To compare the variability in seismic responses of PSCB and MCB, nonlinear time history analyses are also conducted for the MCB. This research concentrates on the displacement responses of pier columns, including both peak and residual displacements. A total of 20 sets of stochastically simulated DMGMs are used as input, scaled to PGAs of 0.3g, 0.6g, and 0.9g to evaluate structural responses under different seismic intensities. In particular, to assess the self-centering capabilities of both PSCB and MCB, nonlinear time history analyses are extended until the bridges reach a fully stationary state. Consequently, the analysis duration is set to 30 s for all cases, even though the simulated seismic motions last only 20 s.

4.1 Influences of soil−structure interaction and seismic motion input types

To explore the effects of SSI and seismic motion input types on the seismic response of the PSCB, four analytical cases are elaborately designed in this study, as illustrated in Tab.2. Specifically, Case 1 serves as a benchmark, accounting for all factors, including SSI and DMGMs. A pile-free FE model is adopted for Case 2, i.e., the bridge structure is considered to be rigidly connected to the ground. Compared with Case 1, Case 3 utilizes the seismic motion input type with uniform excitation, which means that all pile foundations are subjected to identical earthquake loads as Pile 1 (the location of Pile 1 is illustrated in Fig.1 (a)). For Case 4, the seismic excitations at all the soil spring nodes along the piles are kept consistent with those at the top of the pile foundation.

Fig.7 illustrates the typical response time histories for the pier drift ratio (PDR) of the PSCB for each case under different seismic intensities. As shown, the SSI effect and seismic motion input types affect the structural seismic response, including the trend over time and peak displacement responses. Additionally, the PSCB consistently shows smaller residual displacements under all cases, highlighting its excellent self-centering performance. To further reveal the effects of SSI and seismic motion input types on the seismic responses of the PSCB, the mean peak and residual PDRs for each case under different seismic intensities are shown in Fig.8.

As illustrated in Fig.8(a), ignoring the influence of SSI leads to an underestimation of displacement responses of the PSCB, the corresponding variations in the mean peak PDR are 9.93%, 11.40%, and 12.28% for PGA values of 0.3g, 0.6g, and 0.9g, respectively. These phenomena are due to the fact that the considerations of pile foundations and SSI reduce the total stiffness of the bridge, leading to greater structural seismic responses. The fundamental period of the PSCB considering and ignoring SSI effect are 0.91 and 0.82 s, respectively. A comparison of Cases 1 and 3 reveals that neglecting non-uniform excitation underestimates the seismic response of the PSCB, the mean peak PDR are underestimated by 7.54%, 7.02%, and 9.94% under PGAs of 0.3g, 0.6g, and 0.9g, respectively. This may be because the relative displacement is zero between the different pile foundations of bridges when the input type is uniform seismic motions, the out-of-phase vibrations between different supports arising from the spatial effects of earthquake motions are ignored, and thus incorrectly assess the structural seismic response. Moreover, disregarding depth-varying earthquake loads results in higher displacement responses. For PGA values of 0.3g, 0.6g, and 0.9g, the increases in the mean peak PDR are 3.68%, 7.02%, and 8.19%, respectively. This result may be because the seismic motion intensities at the site surface are usually higher due to the amplification effect of the soil layers on seismic propagation, the utilization of surface seismic excitations as input at all of the underground soil springs will exacerbate the vibration of pile foundations, thus finally lead to an overestimation on the seismic response of the PSCB.

In addition, as presented in Fig.8(b), neglecting SSI and non-uniform excitation result in underestimations of the residual displacement response, while ignoring depth-varying ground motions will result in a greater residual PDR. These observations are consistent with the behavior of the peak displacement response, as the residual PDR is closely related to it, thus following similar variation trends. It should be emphasized that, although the residual displacements of PSCBs differ under various cases, the mean residual PDR remains within a relatively small range. This result is attributed to that the prestress tendons in a PSC can provide restoring force and make the pier column almost return to its original position after earthquake excitations.

In summary, it is crucial to acknowledge the importance of considering SSI and DMGMs in the seismic response analysis of PSCBs. Specifically, ignoring the SSI effect underestimates the seismic responses of bridge piers, such underestimates may have a detrimental impact on the performance-based seismic design of PSCBs. Additionally, the use of uniform seismic excitation or neglecting depth-varying ground motions can result in erroneous response predictions for PSCBs, the peak and residual displacement responses of bridge piers may be underestimated or overestimated, ultimately leading to unsafe design and potential risk.

4.2 Comparison of the precast segmental column bridge and monolithic column bridge

To explore the effect of varying pier column types, two types of bridge FE models, one supported by PSCs and the other supported by MCs, are developed, as mentioned earlier. By considering the SSI effect and employing DMGMs as inputs, their displacement responses under different seismic intensities are calculated, as presented in Fig.9. The results demonstrate that the pier column type prominently influences the time history trend of bridge seismic response. This phenomenon may be because the structural characteristics and mechanical properties between PSCs and MCs are significantly distinct, and thus will lead to different response time history trends for bridges supported by various pier types. Moreover, the results also reveal that, as seismic intensity increases, the differences in displacement response between the two bridge types become more pronounced, particularly in residual displacement.

To further illustrate the seismic response differences between the PSCB and MCB, Fig.10 presents the peak and residual PDRs for both bridges under 20 simulated DMGMs. As shown, the PSCB exhibits a greater PDR than the MCB, with this difference becoming more pronounced as seismic intensity increases. Specifically, the mean peak PDR for the PSCB exceeds that of the MCB by 1.04%, 11.42%, and 12.66% under PGAs of 0.3g, 0.6g, and 0.9g, respectively. These differences can be attributed to that the designed MC and PSC in this study have similar initial stiffnesses, coupled with the lower ED capacity of the PSC compared to the MC. When the earthquake intensity is low, two types of bridges remain within the elastic range, resulting in minimal differences in seismic response. However, as the seismic intensity increases and the bridge structures enter nonlinear states, the weaker ED capacity of the PSC results in a larger peak PDR. However, the residual displacement response of the PSCB is significantly lower than that of the MCB, the mean residual PDR differs by 74.54%, 86.78%, and 85.60% at PGAs of 0.3g, 0.6g, and 0.9g, respectively. This result is attributed to the prestressing tendons in the PSC, which provide restoring forces that enable the pier columns to nearly return to their original positions. The above analysis clearly demonstrates that the differences in seismic response between the PSCB and MCB systems and underscores the superiority of bridges supported by PSCs in reducing residual displacement after earthquakes.

5 Fragility analysis of the precast segmental column bridges subjected to depth-varying multi-support ground motion

In seismic fragility analyses of bridge systems, the maximum displacement (MD) response is commonly employed to define structural damage limit states (LSs) [43,44]. However, the significant economic losses resulting from unrecoverable structural permanent deformation after earthquakes have underscored the importance of considering residual deformation (RD) as a critical performance index in addition to the maximum deformation response. Furthermore, given the specific requirement for self-centering capacity in PSCBs, fragility assessments should comprehensively account for both MD and RD. In addition, for comparison, seismic fragility curves for the MCB are also computed.

5.1 Joint fragility function methodology

This research selects peak and residual PDRs as engineering demand parameters (EDPs) and employs the bivariant fragility computation method proposed by Uma et al. [45]. The exceedance probability of LS given the intensity measure (IM) can be expressed as

(9)P\left[\mathrm{L}\mathrm{S}|\mathrm{I}\mathrm{M}\right]=\iint P\left[\mathrm{L}\mathrm{S}|{\mathrm{E}\mathrm{D}\mathrm{P}}_{\mathrm{R}\mathrm{D},\mathrm{M}\mathrm{D}}\right]|\mathrm{d}P\left[{\mathrm{E}\mathrm{D}\mathrm{P}}_{\mathrm{R}\mathrm{D},\mathrm{M}\mathrm{D}}|\mathrm{I}\mathrm{M}\right],

where P\left[\mathrm{L}\mathrm{S}|{\mathrm{E}\mathrm{D}\mathrm{P}}_{\mathrm{R}\mathrm{D},\mathrm{M}\mathrm{D}}\right] is the probability of a specific LS given the EDPs, P\left[{\mathrm{E}\mathrm{D}\mathrm{P}}_{\mathrm{R}\mathrm{D},\mathrm{M}\mathrm{D}}|\mathrm{I}\mathrm{M}\right] represents the probability density function (PDF) of EDPs for a given IM.

The joint distribution of RD (X) and MD (Y) follows a bivariate lognormal distribution, with the joint PDF given by

(10)\begin{array}{rl}& f_{X,Y}\left(x,y\right)\\ =& \frac{0.5}{xy\pi {\zeta }_X{\zeta }_Y\sqrt{1-{{\rho }_{XY}}^2}}\times \mathrm{e}\mathrm{x}\mathrm{p}\{-\frac{0.5}{1-{{\rho }_{XY}}^2}[\frac{{\left(\mathrm{l}\mathrm{o}\mathrm{g}x-{\lambda }_X\right)}^2}{{{\zeta }_X}^2}\\ & -\frac{2{\rho }_{XY}\left(\mathrm{l}\mathrm{o}\mathrm{g}x-{\lambda }_X\right)\left(\mathrm{l}\mathrm{o}\mathrm{g}y-{\lambda }_Y\right)}{{\zeta }_X{\zeta }_Y}+\frac{{\left(\mathrm{l}\mathrm{o}\mathrm{g}y-{\lambda }_Y\right)}^2}{{{\zeta }_Y}^2}]\},\end{array}

where f_{X,Y}\left(x,y\right) is the joint PDF of X and Y; {\lambda }_X and {\lambda }_Y represent the lognormal mean parameters, while {\zeta }_X and {\zeta }_Y denote the standard deviation parameters of variables X and Y, respectively; {\rho }_{XY} refers to the correlation coefficient between variables X and Y.

By performing the double integration over the joint PDF within the range from zero to the corresponding LSs of RD and MD, the damage probability of reaches or exceeds a performance level can be obtained as

(11)P_{\mathrm{L}{\mathrm{S}}_{ij}}=1-{\int }_0^{{MD}_i}{\int }_0^{{RD}_j}{f}_{X,Y}\left(x,y\right)\mathrm{d}x\mathrm{d}y,

where P_{\mathrm{L}{\mathrm{S}}_{ij}} represents the damage probability of a specific limit state; {MD}_i and {RD}_j are the drift limits of MD and RD, respectively.

The LSs for MD are defined using commonly accepted values [46,47]. For RD, tentative values are employed following the previous studies [48–51]. Notably, piers are identified as fail when either the MD or RD exceeds its corresponding limit, thus defining a unified index that combines both peak and residual PDRs is necessary [52], as presented in Tab.3. In addition, the PGA is employed to represent the intensity of simulated DMGMs, which is generally adopted as IM for seismic fragility analysis of girder bridges [43]. Incremental dynamic analysis (IDA) is conducted using 20 sets of simulated DMGMs, with PGA incremented in steps of 0.1g until reaching a maximum value of 1.5g.

5.2 Effects of soil−structure interaction and seismic motion input types

This study considers four cases to investigate the effects of SSI and seismic motion input types on the seismic fragility of the PSCB, as summarized in Tab.2. A total of 20 (DMGMs) × 15 (seismic intensities) × 4 (analytical cases) = 1200 nonlinear analyses are carried out to obtain EDPs of the studied PSCB under different cases. Fig.11 presents the IDA curves and the mean IDA results for both the peak and residual PDRs of each case. Significant differences are observed in the IDA results among the cases. Specifically, the mean peak PDR in Cases 2 and 3 is lower than in Case 1, while in Case 4, it is higher. A similar influence law is observed for the residual PDR, although the differences between cases are less pronounced compared to the peak PDR.

Based on the IDA results and the joint fragility function method, seismic fragility curves of the PSCB for different cases are computed, as presented in Fig.12. As illustrated in Fig.12(a), disregarding the influence of SSI markedly underestimate the seismic damage probability of the PSCB. This is because the structural overall stiffness is decreased when the SSI effect is considered, which in turn results in a higher seismic fragility. In addition, it is evident from Fig.12(b) that the seismic damage probability of bridges is underestimated when uniform ground motions are employed as inputs. These phenomena may be because the assumption of identical earthquake loads on different pile foundations will ignore the complex out-of-phase vibrations of various piers in real seismic events, and this vibration generally results in a large seismic response of bridge structures. Furthermore, as shown in Fig.12(c), if the depth-varying ground motions are not accounted for, the calculated seismic fragility curves of the PSCB will be higher. This is because seismic motions applied to the soil springs exhibit large variations with soil depth, and the intensities of ground motions within the site are typically smaller than those at surface.

To quantify the influences of SSI and seismic motion input types, the fragility median PGAs (i.e., the PGA of corresponding to 50% damage probability) are computed and extracted from Fig.12. Tab.4 demonstrates the results of fragility median PGA values for moderate damage state under different cases. In contrast to Case 1, the seismic median PGA for LS2 shows a 12.45% overestimation when the SSI effect is neglected. When the uniform seismic excitation is employed as the seismic input, the median PGA for LS2 is overestimated by 11.96%. Moreover, ignoring the depth variation of ground motions leads to a 7.89% underestimation of the fragility median PGA for LS2. It can be concluded the SSI and seismic motion input types prominently affect the seismic fragility assessment of the PSCB, ignoring the effects of these factors results in inaccurate seismic fragility curves. Consequently, considering the SSI and DMGMs in seismic performance assessment becomes imperative to ensure reasonable safety design of the PSCB.

5.3 Comparison of the precast segmental column bridge and monolithic column bridge

To compare the seismic fragility differences between the PSCB and MCB, this study also conducts IDA analyses on the MCB under Case 1. Fig.13 illustrates the results of the IDA analysis for the two types of bridges. As shown, the differences in the IDA analysis results between the PSCB and MCB become more significant as the seismic intensity increases. Specifically, when the PGA ranges from 0 to 0.3g, the differences in peak and residual PDRs are not significant. However, as the seismic intensity increases, the PSCB exhibits larger peak displacements and smaller residual displacements compared to the MCB.

Based on the IDA analysis results shown in Fig.13, seismic fragility curves for the MCB are calculated and compared with those of the PSCB. Notably, given the significant differences in peak and residual displacement responses between the PSCB and MCB, this study not only calculates joint fragility curves considering both peak and residual PDRs as EDP parameters, but also computes fragility curves that account for only peak or residual displacement using the traditional IDA fragility method. As shown in Fig.14(a), when peak PDR is used as the EDP, the PSCB typically exhibits higher seismic fragility than the MCB. However, this difference is not significant in the LS1. This is because the PSCB and MCB have comparable initial stiffness in this study, and at lower seismic intensities, the piers generally remain within the elastic range. Therefore, the differences in peak displacement responses at low seismic intensities are minimal. As seismic intensity increases, the piers enter the nonlinear range, the PSCB show larger displacement responses due to their lower ED capacity. Furthermore, as illustrated in Fig.14(b), when the residual PDR is used as the EDP, the PSCB exhibits significantly lower seismic fragility compared to the MCB. These observations can be explained by the fact that the prestressed tendons in the PSC provide a restoring force to the columns, helping the pier columns maintain smaller residual displacements after earthquake excitations.

The aforementioned discussions highlight the differences in seismic fragility between the MCB and PSCB. It should be noted that when peak or residual displacement is used as the EDP, the resulting differences in seismic fragility are not consistent. Therefore, this study will focus on the joint seismic fragility curves that consider both peak and residual displacements (as shown in Fig.14(c)) to assess the differences in seismic fragility between PSCB and MCB. The results reveal that the differences in seismic fragility between the two types of bridges are not consistent under different damage LSs. Specifically, the seismic fragility of the MCB and PSCB is nearly the same in LS1. However, in LS2 and LS3, the PSCB exhibits higher seismic fragility; for example, in LS2, the median fragility PGA of the PSCB (0.811g) is 12.70% lower than that of the MCB (0.914g). Additionally, in LS4, the PSCB demonstrates superior seismic performance due to the lower residual displacement responses. Overall, the seismic fragility of the PSCB and MCB is comparable; however, PSCBs offer significant advantages in earthquake recovery and long-term safety due to their lower residual displacement after higher-intensity earthquake events.

6 Conclusions

In this paper, a versatile seismic performance assessment method for PSCBs by considering the effects of SSI and DMGMs is proposed. The influences of SSI, non-uniform excitation, and depth-varying ground motions on the seismic response and fragility of PSCBs are systematically examined. In addition, for comparison, the seismic performance of a benchmark MCB is also investigated. From the numerical results on this study, the following key conclusions emerge.

1) The proposed seismic performance assessment approach for PSCBs can thoroughly consider the SSI effect and multi-support ground motions at different depths, which ultimately provides a more rational and reliable seismic behavior analysis of such bridges.

2) Compared to the PSCB fixed to the site surface, the one with SSI exhibits higher seismic fragility. Neglecting the influence of SSI can lead to a 12.45% overestimation of the seismic fragility median PGA of the example PSCB.

3) In contrast to the uniform excitation, the example PSCB exhibits higher seismic responses and damage probabilities under the non-uniform seismic excitation. The fragility median PGA is overestimated (up to 11.96%) under the uniform seismic excitation.

4) The depth variation of earthquake loads affects the seismic response and fragility of the PSCB supported by deep pile foundations. Ignoring depth-varying ground motions leads to an underestimation (up to 7.89%) of the fragility median PGA.

5) The seismic performance of the MCB and PSCB demonstrates significant differences. The PSCB exhibits a higher peak displacement but smaller residual displacement compared to the MCB. When considering both peak and residual displacements as EDPs, the difference in fragility median PGA between the two bridges may reach 12.70%.

In performance-based seismic design criteria, peak and residual displacements serve distinct performance objectives. PSCBs, with their self-centering capability, offer significant advantages in reducing residual displacements, which facilitates rapid functional recovery following severe seismic events. However, the segmental joint behavior of PSCBs may result in larger peak displacements. To address the trade-off, it is suggested to employ displacement restrainers, supplemental ED devices, and optimized joint detailing to reduce peak displacement demands while maintaining the self-centering behavior that minimizes residual displacements. The seismic performance assessment approach presented in this study can provide vital technical basis for the seismic design and post-earthquake repair of PSCBs.

References

[1] CulmoM PLordBHuieMBeermanB. Accelerated Bridge Construction: Experience in Design, Fabrication and Erection of Prefabricated Bridge Elements and Systems: Final Manual. FHWA-HIF-12-013. Washington, D.C.: Federal Highway Administration. Office of Bridge Technology, 2011

[2] Billington S L , Barnes R W , Breen J E . A precast segmental substructure system for standard bridges. Precast/Prestressed Concrete Institute Journal, 1999, 44(4): 56–73

[3] Zhang Y , Fan W , Zhai Y , Yuan W . Experimental and numerical investigations on seismic behavior of prefabricated bridge columns with UHPFRC bottom segments. Journal of Bridge Engineering, 2019, 24(8): 04019076

[4] Zhang K , Jia J , Li N , Zhao J , Bai Y . Analytical model for evaluating lateral force capacity of precast concrete-filled steel tube column. Engineering Structures, 2022, 273: 115106

[5] Chou C C , Chang H J , Hewes J T . Two-plastic-hinge and two dimensional finite element models for post-tensioned precast concrete segmental bridge columns. Engineering Structures, 2013, 46: 205–217

[6] Jia J , Zhang K , Wu S , Guo Y , Du X , Wang X . Seismic performance of self-centering precast segmental bridge columns under different lateral loading directions. Engineering Structures, 2020, 221: 111037

[7] Salehi M , Valigura J , Sideris P , Liel A B . Experimental assessment of second-generation hybrid sliding-rocking bridge columns under reversed lateral loading for free and fixed end rotation conditions. Journal of Bridge Engineering, 2021, 26(10): 04021071

[8] Shim C S , Chung C H , Kim H H . Experimental evaluation of seismic performance of precast segmental bridge piers with a circular solid section. Engineering Structures, 2008, 30(12): 3782–3792

[9] Li C , Bi K , Hao H . Seismic performances of precast segmental column under bidirectional earthquake motions: Shake table test and numerical evaluation. Engineering Structures, 2019, 187: 314–328

[10] Li C , Hao H , Bi K . Numerical study on the seismic performance of precast segmental concrete columns under cyclic loading. Engineering Structures, 2017, 148: 373–386

[11] Dawood H , Elgawady M , Hewes J . Factors affecting the seismic behavior of segmental precast bridge columns. Frontiers of Structural and Civil Engineering, 2014, 8(4): 388–398

[12] Mantawy I M , Thonstad T , Sanders D H , Stanton J F , Eberhard M O . Seismic performance of precast, pretensioned, and cast-in-place bridges: Shake table test comparison. Journal of Bridge Engineering, 2016, 21(10): 04016071

[13] Sideris P , Aref A J , Filiatrault A . Large-scale seismic testing of a hybrid sliding-rocking posttensioned segmental bridge system. Journal of Structural Engineering, 2014, 140(6): 04014025

[14] Zhao L , Bi K , Hao H , Li X . Numerical studies on the seismic responses of bridge structures with precast segmental columns. Engineering Structures, 2017, 151: 568–583

[15] Li S , Zhao T , Alam M S , Cheng Z , Wang J . Probabilistic seismic vulnerability and loss assessment of a seismic resistance bridge system with post-tensioning precast segmental ultra-high performance concrete bridge columns. Engineering Structures, 2020, 225: 111321

[16] Upadhyay A , Pantelides C . Fragility-informed seismic design of multi-column bridge bents with post-tensioned concrete columns for accelerated bridge construction. Engineering Structures, 2022, 269: 114807

[17] Li Q , Bi K , Dong H , Lin Y , Han Q , Du X . Numerical studies on the seismic responses of precast segmental columns-supported bridge structures subjected to near-fault ground motions. Advances in Structural Engineering, 2022, 25(12): 2527–2546

[18] Li C , Li H N , Hao H , Bi K , Chen B . Seismic fragility analyses of sea-crossing cable-stayed bridges subjected to multi-support ground motions on offshore sites. Engineering Structures, 2018, 165: 441–456

[19] Ramadan O M , Mehanny S S , Kotb A A M . Assessment of seismic vulnerability of continuous bridges considering soil–structure interaction and wave passage effects. Engineering Structures, 2020, 206: 110161

[20] Zuo H , Bi K , Hao H . Dynamic analyses of operating offshore wind turbines including soil–structure interaction. Engineering Structures, 2018, 157: 42–62

[21] Zhang X , Far H . Effects of dynamic soil–structure interaction on seismic behaviour of high-rise buildings. Bulletin of Earthquake Engineering, 2022, 20(7): 3443–3467

[22] Cavalieri F , Correia A A , Crowley H , Pinho R . Seismic fragility analysis of URM buildings founded on piles: Influence of dynamic soil–structure interaction models. Bulletin of Earthquake Engineering, 2020, 18(9): 4127–4156

[23] Elwardany H , Seleemah A , Jankowski R , El-Khoriby S . Influence of soil–structure interaction on seismic pounding between steel frame buildings considering the effect of infill panels. Bulletin of Earthquake Engineering, 2019, 17(11): 6165–6202

[24] Nakhaei M , Ali Ghannad M . The effect of soil–structure interaction on damage index of buildings. Engineering Structures, 2008, 30(6): 1491–1499

[25] Pan H , Li C , Tian L . Seismic fragility analysis of transmission towers considering effects of soil–structure interaction and depth-varying ground motion inputs. Bulletin of Earthquake Engineering, 2021, 19(11): 4311–4337

[26] Bi K , Hao H . Modelling and simulation of spatially varying earthquake ground motions at sites with varying conditions. Probabilistic Engineering Mechanics, 2012, 29: 92–104

[27] Hao H , Oliveira C , Penzien J . Multiple-station ground motion processing and simulation based on smart-1 array data. Nuclear Engineering and Design, 1989, 111(3): 293–310

[28] Kiureghian A D . A coherency model for spatially varying ground motions. Earthquake Engineering & Structural Dynamics, 1996, 25(1): 99–111

[29] Li C , Diao Y , Li H N , Pan H , Ma R , Han Q , Xing Y . Seismic performance assessment of a sea-crossing cable-stayed bridge system considering soil spatial variability. Reliability Engineering & System Safety, 2023, 235: 109210

[30] Zhao L , Hao H , Bi K , Li X . Numerical study of the seismic responses of precast segmental column bridge under spatially varying ground motions. Journal of Bridge Engineering, 2018, 23(12): 04018096

[31] Li C , Li H , Hao H , Bi K , Tian L . Simulation of multi-support depth-varying earthquake ground motions within heterogeneous onshore and offshore sites. Earthquake Engineering and Engineering Vibration, 2018, 17(3): 475–490

[32] Wang S , Hao H . Effects of random variations of soil properties on site amplification of seismic ground motions. Soil Dynamics and Earthquake Engineering, 2002, 22(7): 551–564

[33] JTG/TB02-01-2008. Guidelines for Seismic Design of Highway Bridges. Beijing: Ministry of Communications of the People’s Republic of China, 2008

[34] MazzoniSMcKennaFScottM HFenvesG L. The Open System for Earthquake Engineering Simulation (OpenSEES) User Command-Language Manual. Berkeley: Pacific Earthquake Engineering Research Center, University of California, Berkeley, 2006

[35] Lee W K , Billington S L . Performance-based earthquake engineering assessment of a self-centering, post-tensioned concrete bridge system. Earthquake Engineering & Structural Dynamics, 2011, 40(8): 887–902

[36] SakaiJMahinS. Analytical Investigations of New Methods for Reducing Residual Displacements of Reinforced Concrete Bridge Columns. PEER Report 2004/02. Berkeley: Pacific Earthquake Engineering Research Center, University of California, Berkeley, 2004

[37] Cai Z K , Wang Z , Yang T . Experimental testing and modeling of precast segmental bridge columns with hybrid normal- and high-strength steel rebars. Construction & Building Materials, 2018, 166: 945–955

[38] APIRP 2A-WSD. Recommended Practice for Planning, Design and Constructing Fixed Offshore Platforms. Washington, D.C.: American Petroleum Institute, 1993

[39] HafezDH. Distributed parameter sensitivity analysis of laterally loaded piles in non-homogeneous soil (soft clay overlying sand under cyclic loading). Dissertation for the Doctoral Degree. Windsor: University of Windsor, 2007

[40] Li C , Hao H , Li H , Bi K , Chen B . Modeling and simulation of spatially correlated ground motions at multiple onshore and offshore sites. Journal of Earthquake Engineering, 2017, 21(3): 359–383

[41] WolfJ. Dynamic Soil–Structure Interaction. New Jersey: Prentice Hall Inc., 1985

[42] Yang J , Sato T . Interpretation of seismic vertical amplification observed at an array site. Bulletin of the Seismological Society of America, 2000, 90(2): 275–285

[43] Ma H B , Zhuo W D , Lavorato D , Nuti C , Fiorentino G , Marano G C , Greco R , Briseghella B . Probabilistic seismic response and uncertainty analysis of continuous bridges under near-fault ground motions. Frontiers of Structural and Civil Engineering, 2019, 13(6): 1510–1519

[44] Tavares D H , Padgett J E , Paultre P . Fragility curves of typical as-built highway bridges in eastern Canada. Engineering Structures, 2012, 40: 107–118

[45] Uma S , Pampanin S , Christopoulos C . Development of probabilistic framework for performance-based seismic assessment of structures considering residual deformations. Journal of Earthquake Engineering, 2010, 14(7): 1092–1111

[46] Kim S H , Shinozuka M . Development of fragility curves of bridges retrofitted by column jacketing. Probabilistic Engineering Mechanics, 2004, 19(1-2): 105–112

[47] Yi J H , Kim S H , Kushiyama S . PDF interpolation technique for seismic fragility analysis of bridges. Engineering Structures, 2007, 29(7): 1312–1322

[48] Shrestha B , Li C , Hao H , Li H . Performance-based seismic assessment of superelastic shape memory alloy-reinforced bridge piers considering residual deformations. Journal of Earthquake Engineering, 2017, 21(7): 1050–1069

[49] Vosooghi A , Saiidi M S . Seismic damage states and response parameters for bridge columns. ACI Special Publications, 2010, 271: 29–46

[50] Dong H , Huang H , Ma R , Du X , Han Q . Quantification of multi-level residual drift ratios of RC bridge piers based on traffic flow capacity and its application in seismic failure assessment of RC bridge piers with EDBs. Bulletin of Earthquake Engineering, 2024, 22(11): 5917–5949

[51] Wang X , Ye A , Yang D , Wu X , Zhou L , Song K , Li J , Peng J , Lou L , Wei X . Component restoration models of highway bridges for resilience assessment: A nationwide expert-opinion survey study and application. Earthquake Spectra, 2025, 41(1): 495–523

[52] Dong H , Li Y , Bi K , Han Q , Du X . Seismic performance assessment of RC bridge piers with variable hysteresis performance dampers. Engineering Structures, 2024, 310: 118085

RIGHTS & PERMISSIONS

Higher Education Press